A Norman window has the shape of a rectangle surmounted by a semicircle. Find the dimensions of a Norman window of perimeter 24 ft that will admit the greatest possible amount of light.
I know that I need to find two different equation to make the main equation have one variable. I would assume in this case we would use the radius of the semicircle to find something in this equation. P=24. I also know that we can split this into two different equation which would be the area of a rectangle and the area of a semicircle. The area of the rectangle is L*w and I have no idea what the area of the semicircle is. I also think that in this case we are trying to maximize the area of the overall shape. Ive been working on this for a while but I keep getting it wrong. I need help!
1 Answer
Let us explore the consequences of making the radius of the semicircle equal to $x$. Let the height of the rectangular part be $h$.
Then the perimeter is equal to $2h+ \pi x+2x$. This is equal to $24$, and therefore $$2h+\pi x+2x=24,\quad\text{or equivalently}\quad h=\frac{1}{2}[24-x(\pi+2)].\tag{1}$$
Note that the area of the window is equal to $2xh+\frac{\pi}{2}x^2$.
Substitute the expression for $h$ that we got in (1) into the formula for the area of the window. We will get a quadratic in $x$, with somewhat messy coefficients. Now maximize in one of the usual ways.
